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J. Math. Anal. Appl. ••• (••••) •••–•••
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www.elsevier.com/locate/jmaa
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Existence of solutions to boundary value problems
for dynamic systems on time scales
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P. Amster , C. Rogers , C.C. Tisdell
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c,∗
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a Departamento de Matemática, Universidad de Buenos Aires, Buenos Aires, Argentina
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b Australian Research Council Centre of Excellence for Mathematics and Statistics of Complex Systems,
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The University of New South Wales, Sydney 2052, Australia
c School of Mathematics, The University of New South Wales, Sydney 2052, Australia
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Received 11 October 2004
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Submitted by J. Henderson
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Abstract
Here, we investigate systems of boundary value problems for dynamic equations on time scales.
Using a generalized relationship between the boundary conditions and a certain subset of the solution
space, the existence of solutions is established through topological arguments. The main tools used
are Leray–Schauder and Brouwer degree theory.
 2004 Published by Elsevier Inc.
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Keywords: Time scales; Boundary value problem; Existence of solutions; Topological methods
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1. Introduction
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The concept of dynamic equations on time scales (aka measure chains) was introduced
in 1990 by Hilger [10] with the motivation of providing a unified approach to continuous
and discrete calculus. Thus, the notion of a generalized derivative y ∆ (t) was introduced,
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* Corresponding author.
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E-mail addresses: [email protected] (P. Amster), [email protected] (C. Rogers),
[email protected] (C.C. Tisdell).
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0022-247X/$ – see front matter  2004 Published by Elsevier Inc.
doi:10.1016/j.jmaa.2004.11.039
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where the domain of the function y(t) is a so-called “time scale” (an arbitrary closed nonempty subset of R). If the time scale is R then the usual derivative is retrieved, that is
y ∆ (t) = y (t). On the other hand, if the time scale is taken to be Z, then the generalized
derivative reduces to the usual forward difference, that is y ∆ (t) = ∆y(t). Recent accounts
of the theory of dynamical equations on time scales has been given by Bohner and Peterson
[5] and Bohner et al. [6].
This paper examines the existence of solutions to the boundary value problem (BVP)
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y ∆ = f t, y σ , t ∈ [a, b]T,
(1.1)
2 ∆
∆
(1.2)
g y(a), y σ (b) ; y (a), y σ (b) = (0, 0),
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P. Amster et al. / J. Math. Anal. Appl. ••• (••••) •••–•••
where f : [a, b]T × Rn → Rn is vector-valued, g : Λ̄ × R2n → R2n with
Λ̄ = y(a), y σ 2 (b) ∈ R2n : y(a) R, y σ 2 (b) R
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for a certain constant R > 0 and where T in (1.1) is a so-called “time scale.”
The essential terminology of time scales is set down below.
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Definition 1.1. A time scale T is a nonempty, closed subset of R.
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A time scale may or may not be connected, whence the concept of jump operation is
necessary.
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Additionally σ (sup T) = sup T, if sup T < ∞, and ρ(inf T) = inf T, if inf T > −∞. Furthermore, denote σ 2 (t) = σ (σ (t)) and y σ (t) = y(σ (t)).
Throughout this work the assumption is made that T has the topology that it inherits
from the standard topology on the real numbers R. It is also assumed throughout that
a < b are points in T with [a, b]T = {t ∈ T: a t b}.
The jump operators σ and ρ allow the classification of points in a time scale in the
following way: if σ (t) > t, then the point t is called right-scattered; while if ρ(t) < t,
then t is termed left-scattered. If t < sup T and σ (t) = t, then the point t is called rightdense; while if t > inf T and ρ(t) = t, then we say t is left-dense.
If T has a left-scattered maximum at m, then we define Tk = T − {m}. Otherwise
k
T = T.
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Definition 1.2. The forward (backward) jump operator σ (t) at t for t < sup T (respectively
ρ(t) at t for t > inf T) is given by
σ (t) = inf{τ > t: τ ∈ T},
ρ(t) = sup{τ < t: τ ∈ T} for all t ∈ T.
Definition 1.3. Fix t ∈ Tk and let y : T → Rn . Then y ∆ (t) is the vector (if it exists) with
the property that given ε > 0 there is a neighbourhood U of t such that, for all s ∈ U and
each i = 1, . . . , n,
yi σ (t) − yi (s) − y ∆ (t) σ (t) − s εσ (t) − s .
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[DTD5] P.2 (1-13)
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Here y ∆ (t) is termed the (delta) derivative of y(t) at t.
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P. Amster et al. / J. Math. Anal. Appl. ••• (••••) •••–•••
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Theorem 1.1. Assume that y : T → Rn and let t ∈ Tk .
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(i) If y is differentiable at t, then y is continuous at t.
(ii) If y is continuous at t and t is right-scattered, then y is differentiable at t with
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y ∆ (t) =
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y ∆ (t) = lim
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y(t) − y(s)
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t −s
(iv) If y is differentiable at t, then y(σ (t)) = y(t) + µ(t)y ∆ (t).
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Definition 1.4. The function y is said to be right-dense continuous, that is y ∈ Crd (T; Rn ),
if at all t ∈ T then:
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(a) y is continuous at every right-dense point t ∈ T, and
(b) lims→t − y(s) exists and is finite at every left-dense point t ∈ T.
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t
y(s)∆s = Y (t) − Y (a).
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Definition 1.5. Let y ∈ Crd . If Y ∆ (t) = y(t), then
Definition 1.6. A function y : [a, b]T → Rn is called a solution to (1.1) if
y ∈ y: y ∈ C a, σ 2 (b) T ; Rn , y ∆∆ ∈ Crd [a, b]T; Rn
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2 ([a, b] ; Rn ).
and y satisfies (1.1) for all t ∈ [a, b]T . We denote this solution space by Crd
T
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A further assumption (as imposed in [8]) is that T is such that [y(σ (t))]∆ 2 exists for
any solution y to (1.1).
The theory of scalar BVPs on time scales has developed at a rapid rate. We cite [1,2,
5–7,9,12,14] and references therein for the main results for the case n = 1.
The theory of BVPs on time scales for systems of equations (n > 1) is yet to be fully
developed with [8] appearing to be the only work on the subject. Here we develop the
existence theory for this wider class of problems.
The paper is organized in the following way.
In Section 2, we introduce and develop “compatibility” theory for the boundary conditions (1.2). The idea involves the set Λ̄ (determined by some constant R > 0) and Brouwer
degree theory. Examples which illustrate the theory are presented.
In Section 3, the compatibility conditions are applied, in conjunction with homotopy
methods (involving Leray–Schauder degree), to formulate existence theorems for solutions to (1.1), (1.2). Additional examples and corollaries illustrating the theorems are then
presented.
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(iii) If y is differentiable and t is right-dense, then
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y(σ (t)) − y(t)
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σ (t) − t
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It is important to note that the new existence results include a wide range of boundary
conditions as special cases, including the Dirichlet boundary conditions
y(a) = A,
y σ 2 (b) = B,
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and the Sturm–Liouville boundary conditions
γ y σ 2 (b) + δy ∆ σ (b) = F,
αy(a) − βy ∆ (a) = E,
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as well as nonlinear variations, such as
y ∆ σ (b) = q y σ 2 (b) .
y ∆ (a) = p y(a) ,
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Of particular interest are the homogeneous Neumann boundary conditions
y ∆ (a) = 0,
y ∆ σ (b) = 0,
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2. Compatibility of boundary conditions
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Definition 2.1. Let R > 0 be a constant. The vector field Ψ = (ψ1 , ψ2 ) ∈
termed strongly inwardly pointing on Λ̄ if ∀ (C, D) ∈ Λ̄ we have
C, ψ1 (C, D) < 0 for C = R,
D, ψ2 (C, D) > 0 for D = R.
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(1.3)
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C(Λ̄; R2n )
is
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(2.1)
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(2.2)
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is continuous and strongly inwardly pointing on Λ̄, where
Λ̄ = (C, D) ∈ R2n : C R, D R
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as (2.1) and (2.2) hold.
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Example 1. For every constant R > 0 the vector field
Ψ (C, D) = ψ1 (C), ψ2 (D) = (−C, D)
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∈ C(Λ̄ × R2n ; R2n )
Definition 2.2. Let R > 0 be a constant. Then g
is said to be strongly
compatible with Λ̄ if, for all strongly inwardly pointing vector fields Ψ = (ψ1 , ψ2 ) ∈
C(Λ̄; R2n ) on Λ̄, we have
g Ψ (C, D) = (0, 0), ∀(C, D) ∈ ∂Λ,
dB g Ψ , Λ, (0, 0) = 0,
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(2.3)
(2.4)
where g Ψ (C, D) = g((C, D); (ψ1 (C, D), ψ2 (C, D))) and dB (g Ψ , Λ, (0, 0)) is the
Brouwer degree of g Ψ at (0,0) relative to Λ.
For more on Brouwer degree, including its calculation and other properties, we refer the
reader to [11, Chapters 1–3].
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since the existence theorems for the BVP (1.1), (1.3) appear to be new in the time scale
setting when n > 1.
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then Ψ above simplifies to
Ψ (C, D) = ψ1 (C), ψ2 (D) .
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For more on compatibility, we refer the reader to [13].
To illustrate the above, we now present some lemmata involving the compatibility of
special cases of (1.2).
Thus, consider the boundary conditions
y ∆ (a) = p y(a) ,
y ∆ σ (b) = q y σ 2 (b) ,
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where p and q map B̄R → Rn (for some known R > 0) and may be nonlinear. Writing
these boundary conditions in the form (1.2), we have
g y(a), y σ 2 (b) ; y ∆ (a), y ∆ σ (b)
= y ∆ (a) − p y(a) , y ∆ σ (b) − q y σ 2 (b)
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= (0, 0).
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(2.5)
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The interest in studying boundary conditions (2.5) is two-fold. Firstly, if p ≡ 0 ≡ q,
then (2.5) become the familiar homogeneous Neumann boundary conditions
y ∆ (a) = 0 = y ∆ σ (b)
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If the boundary conditions are separated, that is
g = g1 y(a), y ∆(a) , g2 y σ 2 (b) , y ∆ σ (b) ,
of classical importance. Secondly, (2.5) arise in applications. For example, (1.1), (2.5)
models the axial deformation of a nonlinear elastic beam with two nonlinear elastic springs
acting at the boundary according to (2.5), with T = R, n = 1 and suitable conditions
imposed on p and q (see [3]).
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Lemma 2.1. Let p, q ∈ C(B̄R ; Rn ) and R > 0 be a constant. If p and q satisfy
C, p(C) 0 for C = R,
D, q(D) 0 for D = R,
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(2.6)
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(2.7)
Proof. The first step is to show that (2.3) holds for all strongly inwardly pointing vector
fields Ψ = (ψ1 , ψ2 ) ∈ C(Λ̄; R2n ).
We write (2.5) in the form (2.3) by replacing: “y(a)” with “C”; “y(σ 2 (b))” with “D”;
∆
“y (a)” with “ψz (C)” and replace “y ∆ (σ (b))” with “ψ2 (D)”, so we obtain
(2.8)
g (C, D); ψ1 (C), ψ2 (D) = ψ1 (C) − p(C), ψ2 (D) − q(D) .
Consider (2.8) for (C, D) ∈ ∂Λ, so C = R and/or D = R.
For C = R we have ψ1 (C) and p(C) satisfying (2.1) and (2.6), respectively, so
0 > C, ψ1 (C) − C, p(C) for C = R,
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then the boundary conditions (2.5) are strongly compatible with Λ̄.
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[DTD5] P.5 (1-13)
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ψ1 (C) − p(C) = 0 for C = R.
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and it needs to be shown that both dB (ψ1 − p, BR , 0) and dB (ψ2 − q, BR , 0) are nonzero.
To compute dB (ψ1 − p, BR , 0), consider the family of mappings
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λ[ψ1 − p] + (1 − λ)(−I )
for λ ∈ [0, 1],
and thus
λ ψ1 (C) − p(C) + (1 − λ)(−C) = 0
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(2.10)
since 0 ∈ BR .
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we can show that
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λ ∈ [0, 1],
dB (ψ2 − q, BR , 0) = 1 (= 0)
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by homotopy methods similar to the case for dB (ψ1 − p, BR , 0). Therefore
dB g Ψ , Λ, (0, 0) = −1 × 1 = 0
and (2.4) holds for all strongly inwardly pointing vector fields Ψ ∈ C(Λ̄; R2n ).
Thus the boundary conditions (2.5) are strongly compatible with Λ̄ and this concludes
the proof. 2
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By considering the family of mappings
λ[ψ2 − q] + (1 − λ)I,
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is defined and (2.9) is a homotopy so that (2.10) is independent of λ. Therefore
dB λ[ψ1 − p] + (1 − λ)(−I ), BR × [0, 1], 0
= dB (−I, BR , 0) = −1 = 0
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for C = R and λ ∈ (0, 1].
If λ = 0 and C = R, then (2.9) becomes −C = 0.
Therefore
dB λ[ψ1 − p] + (1 − λ)(−I ), BR × [0, 1], 0
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(2.9)
where I = identity.
Arguing as in Step 1 of the proof, we see that
0 < C, λ ψ1 (C) − p(C) + (1 − λ)(−C) for C = R and λ ∈ (0, 1]
= dB (ψ1 − p, BR , 0)
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For D = R we have ψ2 (D) and q(D) satisfying (2.2) and (2.7), respectively, so the
inequality 0 < D, ψ2 (D) − D, q(D) leads to ψ2 (D) − q(D) = 0 for D = R similarly
as above.
The second step of the proof is to show that (2.4) holds for all strongly inwardly pointing
Ψ ∈ C(Λ̄; R2n ).
Since (2.3) holds for all (C, D) ∈ ∂Λ, the Brouwer degree (2.4) is defined. To compute (2.4), we use the fact that the boundary conditions (2.5) are separated, so
dB g Ψ , Λ, (0, 0) = dB (ψ1 − p, BR , 0) × dB (ψ2 − q, BR , 0)
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so 0 < ψ1 (C) − p(C) and
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and consequently
0 < C, ψ1 (C) − p(C) Cψ1 (C) − p(C)
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Example 2. Consider the following boundary conditions in the two-dimensional case
(n = 2):
∆ y1 (a)
[y1 (a)]3 − y2 (a)
=
,
(2.11)
y1 (a) + [y2 (a)]3
y2∆ (a)
∆ 2
y1 (σ (b))
y2 (σ 2 (b)) − [y1 (σ 2 (b))]3
=
.
(2.12)
−y1 (σ 2 (b)) − [y2(σ 2 (b))]3
y2∆ (σ 2 (b))
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Let R > 0 be a constant. The boundary conditions (2.11), (2.12) are strongly compatible
with every Λ̄ since all the conditions of Lemma 2.1 are satisfied.
An important corollary to Lemma 2.1 follows in the case p ≡ 0 ≡ q in (2.5). We then
have the homogeneous Neumann boundary conditions
(2.13)
y ∆ (a) = 0 = y ∆ σ (b) .
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= (0, 0).
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Proof. The boundary conditions (2.14) can be written in the form (1.2) as
g y(a), y σ 2 (b) ; y ∆ (a), y ∆ σ (b)
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Lemma 2.3. Let T = 0 and K > 0 be constants. The scalar boundary conditions (2.14)
are strongly compatible with Λ̄ for R = |T |.
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(2.15)
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To show that (2.15) is strongly compatible with Λ̄, we first write (2.15) in the form (2.3)
and so obtain
g Ψ (C, D) = g (C, D); ψ1 (C), ψ2 (D) = ψ1 (C), D 4 + Kψ2 (D) − T 4 , (2.16)
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where Ψ = (ψ1 , ψ2 ) ∈ C(Λ̄; R2 ) and are strongly inwardly pointing vector fields on Λ̄,
with (2.1) and (2.2) holding.
Since n = 1, the inequalities in (2.1) and (2.2) simplify to
ψ1 −|T | > 0,
(2.17)
ψ1 |T | < 0,
ψ2 |T | > 0,
ψ2 −|T | < 0.
(2.18)
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where K > 0 and T are constants. For T = R such boundary conditions arise in the study
of internal heat generation or lateral surface heating of a thin rod occupying the region
[a, b] (see [4]). In this case T is the ambient temperature. 2
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Proof. It is evident that all the conditions of Lemma 2.1 are satisfied with p ≡ 0 ≡ q and
the result follows.
Now consider the nonlinear scalar boundary conditions (n = 1) given by
2 4
y σ (b) + Ky ∆ σ (b) = T 4 ,
(2.14)
y ∆ (a) = 0,
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Corollary 2.2. Let R > 0 be a constant. The Neumann boundary conditions (2.13) are
strongly compatible for every Λ̄.
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The first step is to show that (2.16) = (0, 0) for (C, D) ∈ ∂Λ. See that ψ1 (C) = 0 for
C = ±|T | because (2.17) holds.
For D = |T | we have
D 4 + Kψ2 (D) − T 4 = Kψ2 |T | > 0, because (2.18) holds.
For D = −|T | we have
D 4 + Kψ2 (D) − T 4 = Kψ2 −|T | < 0,
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dB g , Λ, (0, 0) = −1 (= 0)
Ψ
3. Existence of solutions
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Consider the BVP
2
y ∆ = m t, y σ ,
t ∈ [a, b]T,
2 y σ (b) = D,
y(a) = C,
× Rn
where m : [a, b]T
→ Rn
and C, D
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σ
(b)
G(t, s)m s, y σ (s) ∆s + φ(C, D)(t),
y(t) =
a
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where
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G(t, s) =

2(b)−σ (s))
 −(t −a)(σ
,
2
σ (b)−a
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(3.1)
(3.2)
 −(σ (s)−a)(σ 2(b)−t ) ,
σ 2 (b)−a
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Rn ,
then (3.1), (3.2) has a
t ∈ a, σ 2 (b) T ,
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Proof. The BVP (3.1), (3.2) is equivalent to the integral equation [7]
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Theorem 3.1. If m is continuous and bounded on [a, b]T ×
2 ([a, b] ; Rn ) for each given C, D ∈ Rn .
solution y ∈ Crd
T
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∈ Rn .
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and the boundary conditions (2.14) are strongly compatible with Λ̄.
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By continuity of Ψ and by (2.17) and (2.18), we have
dB (ψ1 , B|T | , 0) = −1 and dB D 4 + Kψ2 − T 4 , B|T | , 0 = 1.
Whence
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So (2.16) = (0, 0) for (C, D) ∈ ∂Λ.
The second step of the proof involves showing (2.4) holds.
Using the fact that the boundary conditions (2.14) are separated, we may write
dB g Ψ , Λ, (0, 0) = dB (ψ1 , B|T | , 0) × dB D 4 + Kψ2 − T 4 , B|T | , 0 .
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because (2.18) holds.
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(3.3)
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t s,
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and
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Cσ 2 (b) − Da
+ (D − C)t
.
2
σ (b) − a
2
φ(C, D)(t) =
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σ
(b)
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a
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and consider the family of mappings
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I − Tλ
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G(t, s)m s, y σ (s) ∆s + φ(C, D)(t)
(Tλ y)(t) = λ
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Now
for λ ∈ [0, 1].
Tλ (y) MN + K,
where
σ
(b)
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M=
G(t, s)∆s,
max
t ∈[a,σ 2 (b)]T
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N is the bound on m and
K = max φ(C, D)(t) = max C, D .
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We will show that there exists a y ∈ C([a, σ 2 (b)]T ; Rn ) satisfying (3.3) (which will also
2 ).
be in Crd
Define an operator Tλ by
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t ∈[a,σ 2 (b)]T
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and show that
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y − Tλ (y) = 0
for all y ∈ ∂Ω and all λ ∈ [0, 1].
For y ∈ ∂Ω and all λ ∈ [0, 1] we have
y − Tλ (y) y − Tλ (y) MN + K + 1 − (MN + K) = 1 > 0.
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Since Tλ : Ω̄ × [0, 1] → C([a, σ 2 (b)]T ; Rn ) is a compact map, the following Leray–
Schauder degrees are defined and a homotopy principle is applicable (see [11, Chapter 4],
for more on Leray–Schauder degree):
dLS (I − Tλ , Ω, 0) = dLS (I − T1 , Ω, 0) = dLS (I − T0 , Ω, 0).
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To compute dLS (I − T0 , Ω, 0) it is enough to consider the homotopy
I − θ T0 ,
θ ∈ [0, 1],
and it readily follows that dLS (I − T0 , Ω, 0) = 1.
2.
By elementary methods, the solution y ∈ Ω is also in Crd
This concludes the proof. 2
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Now we consider the set
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Theorem 3.2. Let R > 0 be a constant. If f is continuous and satisfies
σ σ y , f t, y > 0 for all t ∈ [a, b]T and y σ = R,
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It is evident that m is continuous and bounded on [a, b]T ×
solution by Theorem 3.1.
Since C, D R we have
y(a), y σ 2 (b) R.
for
y σ R
= Ry σ /y σ = p where p
y(t) < R on a, σ 2 (b) .
T
Now consider (1.1), (1.2).
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(3.5)
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(3.6)
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and by [8, Lemma 3] we have
Proof. Consider the modified BVP
2
y ∆ = m t, y σ , t ∈ [a, b]T,
y(a) = C,
y σ 2 (b) = D,
g (C, D); y ∆ (a), y ∆ σ (b) = (0, 0),
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× Rn
and satisfy
Theorem 3.3. Let R > 0 be a constant. Let f be continuous on [a, b]T
(3.4). If g is continuous on Λ̄ × R2n and strongly compatible with Λ̄, then (1.1), (1.2) has
a solution y with y < R.
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(3.7)
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(3.8)
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(3.9)
where m is given in Theorem 3.2. The latter shows that the problem (3.7), (3.8) has a
2 with y(t) < R on (a, σ 2 (b)) with y(a) R, y(σ 2 (b)) R for
solution y ∈ Crd
T
each C, D R.
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Hence solutions to the BVP (3.5), (3.6) are solutions to the BVP (3.1), (3.2) and this concludes the proof. 2
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Rn
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To show that y(t) < R on
T , it is noted that
σ σ y , m t, y = p, f (t, p) > 0,
(a, σ 2 (b))
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Proof. Consider the modified BVP
2
y ∆ = m t, y σ , t ∈ [a, b]T,
R
σ
σ
for y σ R,
σ f (t, Ry /y )
= y for y σ R,
f (t, y σ ),
2 y(a) = C,
y σ (b) = D.
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(3.4)
then for each C, D ∈ Rn satisfying C, D R the BVP (1.1), (3.2) has a solution y
with
y(t) < R on a, σ 2 (b) ,
2 T
y σ (b) R.
y(a) R,
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It remains to establish that there exists (y, C, D) satisfying (3.7)–(3.9).
It is readily seen that
(I − T1 )(y), g (C, D); y ∆ (a), y ∆ σ (b)
= (0, 0, 0)
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is equivalent to (3.7)–(3.9), so we need to show that the following Leray–Schauder degree
dLS (I − T1 , g), Ω × Λ, (0, 0, 0) = 0,
5
where Tλ is defined in Theorem 3.1 and
Ω = y ∈ C a, σ 2 (b) T ; Rn : y < R .
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Consider the family of functions given by
(I − Tλ )(y), g (C, D); λ y ∆ (a), y ∆ σ (b)
+ (1 − λ) ψ1 (C, D), ψ2 (C, D) = (0, 0, 0),
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(3.10)
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Now consider
C, λy ∆ (a) + (1 − λ)ψ1 (C, D) .
T.
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for y(a) = C = R or
with y < R on
Assume y(a) = C = R and y < R on (a, σ 2 (b))T . Therefore
y(a), y ∆(a) < 0, by [8].
(a, σ 2 (b))
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y(σ 2 (b)) = D = R
Expanding the latter and using the fact that Ψ is strongly inwardly pointing on Λ̄, we
obtain
λ C, y ∆ (a) + (1 − λ) C, ψ1 (C, D) < 0 for all λ ∈ [0, 1].
Similarly, if y(σ 2 (b)) = D = R and y < R on (a, σ 2 (b))T , we have
2 ∆
y σ (b) , y σ (b) > 0.
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A similar argument yields
λ D, y ∆ σ (b) + (1 − λ) D, ψ2 (C, D) > 0
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for all λ ∈ [0, 1].
Therefore (3.10) is nonzero for (C, D) ∈ ∂Λ.
Hence the following degrees are defined and by homotopy methods:
dLS I − Tλ , gλΨ , Ω × Λ, (0, 0, 0)
= dLS I − T1 , g1Ψ , Ω × Λ, (0, 0, 0)
= dLS I − T0 , g0Ψ , Ω × Λ, (0, 0, 0)
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for λ ∈ [0, 1], and Ψ = (ψ1 , ψ2 ) are strongly inwardly pointing vector fields on Λ̄.
In view of the conclusions of Theorems 3.1 and 3.2, we need to show that, for all λ ∈
[0, 1],
gλΨ = g (C, D); λ y ∆ (a), y ∆ σ (b) + (1 − λ) ψ1 (C, D), ψ2 (C, D) = (0, 0),
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= dLS I − T0 , g Ψ , Ω × Λ, (0, 0, 0)
= 1 × dB g Ψ , Λ, (0, 0)
= 0
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by (2.4).
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Thus, there is a solution (y, C, D) to (3.7)–(3.9) for all λ ∈ [0, 1] and for λ = 1, this is a
solution to (1.1), (1.2). This concludes the proof. 2
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Corollary 3.4. Consider the BVP (1.1), (2.5). Let f be continuous and satisfy (3.4). In
addition, let p and q satisfy the conditions of Lemma 2.1. Then the BVP (1.1), (2.5) has a
solution.
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Acknowledgments
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C.C. Tisdell gratefully acknowledges the research support of the Australian Research Council’s Discovery
Projects (DP0450752). P. Amster thanks Fundacion Antorchas for research support.
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References
[1] R.P. Agarwal, D. O’Regan, Nonlinear boundary value problems on time scales, Nonlinear Anal. 44 (2001)
527–535.
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(3.11)
if we choose R = 2.
By Lemma 2.1 and Example 2, the boundary conditions are strongly compatible for any
R > 0, so for the choice R = 2 the boundary conditions are strongly compatible.
All of the conditions of Theorem 3.3 are satisfied and the existence of a solution follows.
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subject to the boundary conditions (2.11), (2.12). We claim that the BVP (3.11), (2.11),
(2.12) has a (nontrivial) solution. Note that f is continuous. Consider, for y = R,
y, f (t, y) = y12 + y22 + y1 = R 2 + y1 > 0,
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Example 3. Consider the system of equations
2 y1∆
y1 − y1 y22 + 1
=
, t ∈ [a, b]T,
2
y2 + y2 y12
y2∆
RR
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Proof. By Corollary 2.2, the boundary conditions (2.13) are strongly compatible with Λ̄.
All of the conditions of Theorem 3.3 are satisfied and the results follows. This concludes
the proof. 2
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Corollary 3.5. Consider the BVP (1.1), (2.13). Let f be continuous and satisfy (3.4). Then
the BVP (1.1), (2.13) has a solution.
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Proof. By Lemma 2.1, the boundary conditions (2.5) are strongly compatible with Λ̄. All
of the conditions of Theorem 3.3 are satisfied and the result follows. This concludes the
proof. 2
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Appl. 266 (2002) 160–168.
[4] L.E. Bobisud, D. O’Regan, Existence of positive solutions for singular ordinary differential equations with
nonlinear boundary conditions, Proc. Amer. Math. Soc. 124 (1996) 2081–2087.
[5] M. Bohner, A. Peterson, Dynamic Equations on Time Scales. An Introduction with Applications, Birkhäuser
Boston, Boston, MA, 2001.
[6] M. Bohner, G. Guseinov, A. Peterson, Introduction to the time scales calculus, in: Advances in Dynamic
Equations on Time Scales, Birkhäuser Boston, Boston, MA, 2003, pp. 1–15.
[7] L. Erbe, A. Peterson, Green’s functions and comparison theorems for differential equations on measure
chains, Dynam. Contin. Discrete Impuls. Systems 6 (1999) 121–137.
[8] J. Henderson, A. Peterson, C.C. Tisdell, On the existence and uniqueness of solutions to boundary value
problems on time scales, Adv. Differential Equations 2004 (2004) 93–109.
[9] J. Henderson, C.C. Tisdell, Topological transversality and boundary value problems on time scales, J. Math.
Anal. Appl. 289 (2004) 110–125.
[10] S. Hilger, Analysis on measure chains—a unified approach to continuous and discrete calculus, Results
Math. 18 (1990) 18–56.
[11] N.G. Lloyd, Degree Theory, Cambridge Tracts in Math., vol. 73, Cambridge Univ. Press, Cambridge, 1978.
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Discrete Contin. Dynam. Systems, in press.
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